Australia retained the Trans-Tasman trophy even though rain intervened to deny them victory in a one-sided second Test, which ended in a draw in Wellington on Tuesday.
New Zealand suffered a heavy nine-wicket defeat in the series opener in Christchurch and McMillan has paid the price for some indifferent performances by New Zealand's batsmen.
New Zealand Cricket announced a reshuffle of the batting order for the third Test, which begins in Auckland on Saturday.
James Marshall will open the batting in his Test debut with Craig Cumming against Australia's feared pace attack of Glenn McGrath, Jason Gillespie and Michael Kasprowicz.
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Fleming fell lbw to McGrath twice in the second Test for a duck and one and is averaging 9.0 from four innings in the series.
Marshall's twin brother Hamish, who scored his maiden Test century in the first Test, remains at number three in the order.
Lou Vincent moves to number five in the batting lineup and Nathan Astle to six.
"James is one of a few players who are adding depth to the pool of players we select from," New Zealand coach John Bracewell said in a statement.
New Zealand were forced to follow on in the second Test as they chased Australia's massive first innings total of 570 for eight declared.
Fleming's side were 48 for three, needing another 278 runs to avoid an innings defeat, when rain stopped play.
New Zealand:
Stephen Fleming (captain), Nathan Astle, Craig Cumming, James Franklin, Hamish Marshall, James Marshall, Chris Martin, Brendon McCullum, Iain O'Brien, Daniel Vettori, Lou Vincent, Paul Wiseman (12th man to be named).
